The electrical conductance of electrode and electrolyte materials can be critical parameter for a variety of devices, including lithium-based batteries, supercapacitors and certain fuel cells. Contact measurements fail to accurately determine this parameter for powder-like materials as the electrical resistance of the grains chain to the current that flows in succession through them is mainly determined by the contact resistances in places the grains contact each other, while these resistances substantially exceed their inner resistances. This is explained by two reasons. First, the contacting grains, when the powder is of minor density (within the operating range of the powder densities being used for electrode coats of the chemical power sources) obtained by vibration compacting, have relatively small areas of contact there among. Second, the surface resistance of the powder grains is as a rule substantially higher of the volume resistance. This is explained by a lower concentration of free charge carriers on the surface of the grains and by their lower mobility, by forming, due to the chemical interaction with the atmosphere, of various types of insulating films, surface layers, inclusions.
The eddy-current method for measuring electrical conductivity is well known and relatively straightforward non-contact method for characterizing the electrical conductivity of continuous media, such as metals and semiconductors. The theory of interaction between the eddy magnetic fields and homogenous conducting materials has been adequately developed, and instruments for electrical conductivity measurement have been produced. However, layers comprising a variety of devices, including lithium-based batteries generally utilize powder like materials, rather than continuous media. Accordingly, the eddy-current measuring method of powder conductivity and the eddy-current method of conductivity measurement for continuous media are necessarily significantly different, as described below.
In contrast to eddy currents in continuous materials, the eddy currents in powders are mainly locked within each separate particle (each grain) of the powder. The active resistance introduced into the eddy current sensor and formed due to the loss of the field source power as a result of the eddy current flowing through the conducting medium is formed in the case of the powder as a sum of the joule losses in each grain.
The density of the eddy current induced by the field in each powder grain, according to the Ohm's law equals:j=σ0E,  (a)where σ0 is the specific electrical conductance of the grain material, E is the electric field intensity in the place of the grain location.
The electric field intensity E is related to the magnetic field intensity of sensor H through the Maxwell's equation:
                              rotE          =                                    -                              μμ                0                                      ⁢                                          ∂                H                                            ∂                t                                                    ,                            (        b        )            where μ is the magnetic permeability of the medium, μ0=4π·10−7H·m−1−magnetic constant.
By introducing vector—potential A: rotA=−μμ0H for the harmonic field of the sensor in the form of a cylindrical inductance coil the following is obtained:Ė=−iωW{dot over (A)},  (c)where i=√{square root over (−1)}, ω−angular frequency, W—number of turns in the sensor.
In a homogeneous magnetic field of a cylindrical inductance coil, the filed intensity H being identical at all points of the inner space of the coil, the eddy current density j induced in the grains of the powder filling this space will be identical too. This follows from equations (a)-(c).
The power of active losses in a single grain of the powder equals:Pn=In2Rn,  (d)where In and Rn—respectively is the eddy current flowing along a circular trajectory in the grain, and the ohmic resistance of the grain material.
A case is now considered where the inner space of height h in a cylindrical inductance coil (an eddy-current sensor) is filled with powder grains of identical size shaped as globes of diameter D and densely arranged relatively to each other.
Then the total power of the active (Joule) losses for all the grains equalsPΣ=N1σ0E2Sl,  (e)where N1 is the number of grains in the inner space of the sensor, S and l correspondingly is the cross-section and length of the eddy current pipe in the grain.
The number of grains N1 is now determined in the first approximation as the relation between the volume of the inner space of the sensor densely filled with the grains and the volume of one grain:
                                          N            1                    =                                    3              2                        ⁢                                                            D                  1                  2                                ⁢                h                                            D                3                                                    ,                            (        f        )            where D1is the inner diameter of the cylindrical inductance coil, h—the height of its inner space, D—grain diameter.
FIG. 1(a) shows the arrangement of the current trajectories in a grain that is found in a homogeneous eddy magnetic field of intensity H. At high frequencies the eddy currents are pressed to the surface of the grains. This phenomenon is known as the skin-effect. The radius of the average (resultant) eddy current pipe in a grain are now determined. To this end, N eddy current pipes found within the region y>0 is considered as shown in FIG. 1(b) which defines coordinates of eddy current pipes. The width of each pipe equals D/2N. The coordinates of the first pipe along the Y-axis: 0, D/2N; of the second pipe D/2N, 2D/2N; of the third pipe 2D/2N, 3D/2N; and so on. The radius of the first eddy current pipe: R1=D/2. The radius of the second pipe is found from the equation of a circle: X2+y2=D2/4; in this case R2=(D2/4−D2/4N2)1/2. The radius of the third pipe: R3=(D2/4−4D2/4N2)1/2; of the fourth pipe: R4=(D2/4−9D2/4N2)1/2, and so on. The radius of the last N-th pipe RN=[D2/4−(N−1)2D2/4N2]1/2. Hence the average radius of an eddy current pipe in a powder grain of globular shape equals:
            R      av        =                  1        N            ⁡              [                                                                              D                  2                                +                                                      D                                          2                      ⁢                      N                                                        ⁢                                                            (                                                                        N                                                                                                                                            ⁢                            2                                                                          -                        1                                            )                                                              1                      /                      2                                                                      +                                                      D                                          2                      ⁢                      N                                                        ⁢                                                            (                                                                        N                                                                                                                                            ⁢                            2                                                                          -                        4                                            )                                                              1                      /                      2                                                                      +                                                                                                                              D                                          2                      ⁢                      N                                                        ⁢                                                            (                                                                        N                                                                                                                                            ⁢                            2                                                                          -                        9                                            )                                                              1                      /                      2                                                                      +                …                +                                                      D                                          2                      ⁢                      N                                                        ⁢                                                            (                                                                        2                          ⁢                          N                                                -                        1                                            )                                                              1                      /                      2                                                                                                          ]              ,or in a general form
                              R          av                =                              D                          2              ⁢                              N                2                                              ⁢                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                                  ⁢                                                            (                                                            N                                                                                                                        ⁢                        2                                                              -                                          n                      2                                                        )                                                  1                  /                  2                                            .                                                          (        g        )            
Hence the average length of an eddy current pipe l=2πRav, while the pipe cross-section is assumed to be equal S=πd0/4. The total number of such pipes in a grain is D/d0.
                              P          Σ                =                                                            3                ⁢                                  π                  2                                ⁢                                  D                  1                                ⁢                                  hd                  0                                ⁢                                  σ                  0                                ⁢                                  E                  2                                                            8                ⁢                D                                      ·                          1                              N                                                                                          ⁢                  2                                                              ⁢                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                                  ⁢                                                            (                                                            N                                                                                                                        ⁢                        2                                                              -                                          n                      2                                                        )                                                  1                  /                  2                                            .                                                          (        h        )            
                              P          Σ                =                                            3.05              ⁢                                                          ⁢                              D                1                2                            ⁢                                                          ⁢                              hd                0                            ⁢                              σ                0                            ⁢                              E                2                                      D                    .                                    (        i        )            
As can be seen from equations (h) and (i), the total sensor field power of the Joule losses in the powder grains that densely fill the inner space of the sensor is a function of the grains diameter.
The influence of the powder density is now considered. The powder density ρp equals:
                                          ρ            p                    =                                    M              V                        =                                                                                                      N                      ρ                                        ·                                          ρ                      0                                                        ⁢                                      V                    0                                                  V                            =                                                N                  ρ                                ·                                  ρ                  0                                ·                                                      2                    ⁢                                          D                      3                                                                            3                    ⁢                                          D                      1                      2                                        ⁢                    h                                                                                      ,                            (        j        )            where M is the powder mass in the inner space of the sensor of volume V, N92 is the number of the powder grains at the specified density, ρ0—grain density, V0—grain volume, D—grain diameter, D1—inner diameter of the cylindrical inductance coil (of the eddy-current sensor), h—height of the sensor inner space of volume V.
Assuming, similar to (i) N=10, (see FIG. 1(b), the expression (e) for the total power of the Joule losses the following is obtained:PΣ=Nρ·0,206π2Dd02σ0E2.  (k)
Determining Nρ from (j) and substituting it into (k), results in:
                              P          Σ                =                                            3              ⁢                              .                            ⁢              05              ⁢                                                          ⁢                              D                1                2                            ⁢                                                          ⁢                              hd                0                            ⁢                              ρ                p                            ⁢                              σ                0                            ⁢                              E                2                                                    D              ⁢                                                          ⁢                              ρ                0                                              .                                    (        l        )            Taking into account that the power of the Joule losses of the sensor field in a conducting medium when eddy currents flow therein, is proportional to the active resistance introduced into the sensor, the following is obtained:Rad(p)=kqρpσ0,  (m)where Rad(p) is the introduced into the sensor active resistance formed during the non-contact control of the powder of density ρp; σ0 is the specific electric conductance of the powder grain; k is the proportionality coefficient, q=3D12hd0E2/Dρ0.
Thus, for the given powder with fixed values of σ0, ρ0 and D the value of the introduced active resistance of the sensor is proportional to the powder density ρp.
The theory of interaction between the axi-symmetrical eddy magnetic field of a cylindrical inductance coil (pass-through eddy-current sensor) and the conducting medium filling the inner space of a sensor has been developed for continuous homogeneous media.
For such a homogeneous medium let us take as an elementary the eddy current pipe of section d0 and diameter D0=2Rav=0,826D (Eqs. g-i). The number of such elementary pipes in the medium filling the inner space of the sensor of height h equals:
                              N          S                =                                                            D                1                2                            ⁢              h                                      0.682              ⁢                                                          ⁢                              D                2                            ⁢                              d                0                                              .                                    (        n        )            The power of the Joule losses in the medium is determined as follows:
                                          P            Σ                          (              S              )                                =                                                    2.98                ·                                                                  ⁢                                  D                  1                  2                                            ⁢                                                          ⁢                              hd                0                            ⁢              σ              ⁢                                                          ⁢                              E                2                                      D                          ,                            (        o        )            where σ is the specific electric conductance of a continuous medium.
Taking into account that the introduced active resistance is Rad(S)=kPΣ(S), from the condition of equality of the introduced active resistances for a continuous medium and for the powder: Rad(P)=Rad(S), on the basis of equations (l) and (o) the following is obtained:
                                                                        3.05                ⁢                                  D                  1                  2                                ⁢                                  hd                  0                                ⁢                                  ρ                  p                                ⁢                                  σ                  0                                ⁢                                  E                  2                                                            D                ⁢                                                                  ⁢                                  ρ                  0                                                      =                                          2                ⁢                                  .                                ⁢                98                ⁢                                  D                  1                  2                                ⁢                                  hd                  0                                ⁢                σ                ⁢                                                                  ⁢                                  E                  2                                            D                                ;                ⁢                                  ⁢        or                            (        p        )                                                      σ            0                                ρ            0                          =                  0          ⁢                      .                    ⁢          977          ⁢                                    σ                              ρ                p                                      .                                              (        q        )            
Taking into account that during the vibration compacting the ratio between the powder grain density and the powder density ρ0/ρp is always above unity, the specific conductance of the powder grains always exceeds the value of the equivalent conductance of continuous homogeneous medium.
The equivalent conductance value can be calculated according to the measured value of the introduced active resistance for powder Rad(P) using the tables contained. An example of such calculation is given in the description of the present invention.
It should be noted that equation (q) does not include the grain diameter D.
The ratio between the specific electric conductance of the powder grain σ0 and the grain density ρ0 is now designated by symbol γ:
                              γ          =                                    σ              0                                      ρ              0                                      ,                            (        r        )            and termed the reduced conductance of the powder grains.
At constant ρ0 value γ is proportional to σ0—the specific electric conductance of the powder grains.
The powder grains are porous. The higher is the porosity, the less is the density of the grains and higher γ. In electrolyte applications for batteries, the electrolyte penetrates into the pores of the coat thus increasing the total surface area of the contact between the electrode coat and the electrolyte (the active surface area of the coat). Therefore, powders featuring a substantial electric conductivity per grain density unit (high γ) are of great value for electrode coats of chemical power sources.
The cathode coat of chemical power sources is of composite structure comprised of the basic powder-like material whose grains feature ionic conductivity, such as spinels LiMn2O4 or MnO2, and of fillers, such as graphite powders and soot. The mixture of such powders in a liquid medium of an organic binder is thoroughly mixed in order to obtain a homogenous mass. In such case a high uniformity of the obtained material, that is the uniformity of the spatial distribution of grains of all the three types of powders is a function of the closeness of the density values of their grains. The lightest component of the mix is soot. Therefore of critical importance is to adequately select the powders of the spinel and graphite of low density of the grains and (naturally) of grains with a high specific conductance, that is, with a high γ value.
The directly proportional dependence between the equivalent σ and density ρp for the given powder with the fixed σ0 and ρ0 values can be disrupted due to the interaction of the magnetic fields of the eddy currents in adjacent grains and the shifting currents among the grains.
The highest effect of this can be observed at maximum compaction when the powder grains are found at the maximum distance from each other.
The intensity of the displacement current is proportional to frequency ω and capacitance C among the particles that increases during the powder compaction.
The mutual inductance of the two circular currents with identical radii Rav (Eq. g), that are flowing in adjacent powder grains can be calculated using the following formula:
                              M          =                                    π              16                        ⁢                                          μ                0                            ⁡                              (                                  d                  a                                )                                      ⁢                                                  ⁢                          ξ              [                                                          ⁢                                                b                  y                                -                1                +                                                                            ξ                      2                                        2                                    ⁢                                      (                                          1                      -                                                                        3                          2                                                ⁢                                                  η                          2                                                                    +                                              1                                                  2                          ⁢                                                                                                          ⁢                                                      η                            3                                                                                                                )                                                  -                                                      5                    8                                    ⁢                                                            ξ                      4                                        ⁡                                          (                                              1                        -                                                  5                          ⁢                                                                                                          ⁢                                                      η                            2                                                                          +                                                                              35                            8                                                    ⁢                                                      η                            4                                                                          -                                                                              3                            8                                                    ·                                                      1                                                          η                              5                                                                                                                          )                                                                      +                                                      35                    32                                    ⁢                                                            ξ                      6                                        ⁡                                          (                                              1                        -                                                                              21                            2                                                    ⁢                                                      η                            2                                                                          +                                                                              189                            8                                                    ⁢                                                      η                            4                                                                          -                                                                              231                            16                                                    ⁢                                                      η                            6                                                                          +                                                  5                                                      16                            ⁢                                                                                                                  ⁢                                                          η                              7                                                                                                                          )                                                                      +                …                            ⁢                                                          ]                                      ,                            (        s        )            where d=2Rav is the circular trajectory diameter of the eddy current; a=D/20 (similar to i), where D is the powder grain diameter; y—distance between the grain centers; b=√{square root over (y2+a2)}, ξ=d/2, η=y/b.
The values of M/M0 calculated according to (s) where M0=M(y=D), that is when two grains are closely arranged, depending on the least distance between the lateral surfaces of the grains x=y−D are given in Table A.
TABLE AM/M0x/D100.9060.10.8350.20.7120.40.6260.60.5540.80.501.0
It follows from Table A that at x=D the mutual inductance between the eddy currents induced in the adjacent powder grains becomes twice lower.
When there is no directly proportional relationship between σ and ρp due to the interaction of the magnetic fields of the grain eddy currents and to the presence of the displacement currents among the grains it is necessary to approximate the equivalent electrical conductance σ as a function of density ρp using a polynomial function of moderate order with a subsequent segregation of the linear term whose coefficient represents a derivative of the equivalent electrical conductance per powder density and is equal to the specific powder grains conductance reduced to their unit density.